Prime matrix rings

Question

I have to answer the following question:

Let $R$ be a unital ring and $n\geq1$ a natural number.

$i)$ Show that $M_{n}(R)$ is prime if and only if $R$ is prime.

$ii)$ Show that $M_{n}(R)$ is semiprime if and only if $R$ is semiprime.

This seems to be intuitive but as so often the case in ring theory, the answers are difficult to formulate.

Can anyone give me some steps in the right direction?

Answer 1

Suppose $M_n(R)$ is a prime ring, and let $a$ and $b$ be elements of $R$. Let $\phi : R \rightarrow M_n(R)$ be the map that sends an element $r$ to the matrix whose 1,1 entry is $r$ and whose other entries are 0.

If $arb=0$ for all $r$ in $R$, then $\phi(a) \phi(r) \phi(b)$ is the zero matrix. And if $M_n(R)$ is a prime ring, it follows that either $\phi(a)$ of $\phi(b)$ must be zero. But therefore either $a$ or $b$ must be zero, showing that $R$ is a prime ring.

Conversely, suppose that $R$ is a prime ring, and let $A$ and $B$ be elements of $M_n(R)$. If $ARB=0$ for all elements $R$ in $M_n(R)$, then certainly $A S \phi(r) T B = 0$ for every $r$ in R and every pair of permutation matrices $S$ and $T$. Choosing $S$ and $T$ appropriately, we can obtain any of the $A_{ij} r B_{kl}$ as the 1,1 entry. If some particular $A_{ij}$ is nonzero and $R$ is prime, it follows that $B$ must be the zero matrix. But therefore $M_n(R)$ is a prime ring.

Answer 2

These are both trivial consequences of the last question and answer you got on the topic:

1. $\{0\}$ is prime in $R$ iff $M_n(\{0\})$ is prime in $M_n(R)$.

2. $S\lhd R$ is an intersection of primes in $R$ iff $M_n(S)$ is an intersection of primes in $M_n(R)$. In particular the zero ideal in $R$ is semiprime iff the zero ideal in $M_n(R)$ is.